wireless network pricing chapter 5: monopoly and price...

Wireless Network PricingChapter 5: Monopoly and Price Discriminations

Jianwei Huang & Lin Gao

Network Communications and Economics Lab (NCEL)Information Engineering Department

The Chinese University of Hong Kong

Huang & Gao ( c©NCEL) Wireless Network Pricing: Chapter 5 September 28, 2016 1 / 73

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Chapter 5: Monopoly and Price Discriminations


Focus of This Chapter

Key Focus: This chapter focuses on the problem of profitmaximization in a monopoly market, where one service provider(monopolist) dominates the market and seeks to maximize its profit.

Theoretic Approach: Price TheoryI Price theory mainly refers to the study of how prices are decided and

how they go up and down because of economic forces such as changesin supply and demand (from Cambridge Business English Dictionary)


Price Theory

Mainly follow the discussions in “Price Theory and Applications” byB. Peter Pashigian (1995) and Steven E. Landsburg (2010)

Part I: Monopoly PricingI The service provider charges a single optimized price to all the

consumers.

Part II: Price DiscriminationI The service provider charges different prices for different units of

products or to different consumers.


Section 5.1Theory: Monopoly Pricing


What is Monopoly?

Etymology suggests that a “monopoly” is a single seller, i.e., the onlyfirm in its industry.

I Question: Is Apple a monopoly?F It is the only firm that sells iPhone;F It is not the only firm that sells smartphones.

The formal definition of monopoly is based on the monopoly power.

Definition (Monopoly)

A firm with monopoly power is referred to as a monopoly or monopolist.


What is Monopoly Power?

Monopoly power (or market power) is the ability of a firm to affectmarket prices through its actions.

Definition (Monopoly Power)

A firm has monopoly power, if and only if

(i) it faces a downward-sloping demand curve for its product, and

(ii) it has no supply curve.

(i) implies that a monopolist is not perfectly competitive. That is, he isable to set the market price so as to shape the demand.

(ii) implies that the market price is a consequence of the monopolist’sactions, rather than a condition that he must react to.


Profit Maximization Problem

P: the market price that a monopolist chooses;

Q , D(P): the downward-sloping demand curve that the monopolistfaces;

Definition (Monopolist’s Profit Maximization Problem)

The monopolist’s choice of market price P to maximize his profit (revenue)

π(P) , P · Q = P · D(P).

Here we tentatively assume that there is no production cost, henceprofit = revenue.

In general, profit = revenue - cost.


Profit Maximization Problem

The first-order condition:

dπ(P)

dP= Q + P · dQ

dP= 0

The optimality condition:

P · 4Q

Q · 4P+ 1 = 0

I 4P is a very small change in price, and 4Q is the correspondingchange in demand quantity.


Demand Elasticity

Price Elasticity of Demand (defined in Section 3.2.5)

η ,4Q/Q

4P/P=

P · 4Q

Q · 4P

I The ratio between the percentage change of demand and thepercentage change of price.

A Closely Related Question: Under a particular price P and demandQ = D(P), how much should the monopolist lower his price to selladditional 4Q units of product?

⇒ Answer:

4P =P

Q · η· 4Q


Demand Elasticity

The monopolist’s total profit change by selling additional 4Q units ofproduct:

4π , P · 4Q − |4P| · Q

= P · 4Q −∣∣∣∣ P

Q · η· 4Q

∣∣∣∣ · Q= P · 4Q ·

(1− 1

|η|

)I P · 4Q is the profit gain that the monopolist achieves, by selling

additional 4Q units of product at price P;I |4P| · Q is the profit loss that the monopolist suffers, due to the

decrease of price (by |4P|) for the previous Q units of product.I We ignore the higher order term of 4P · 4Q.


Demand Elasticity

Monopolist’s Total Profit Change:

4π = P · 4Q ·(

1− 1

|η|

)I If |η| > 1, then 4π > 0. This implies that the monopolist has incentive

to decrease the price when |η| > 1.I If |η| < 1, then 4π < 0. This implies that the monopolist has incentive

to increase the price when |η| < 1.I If |η| = 1, then 4π = 0. This implies that the monopolist has no

incentive to increase or decrease the price when |η| = 1 (assuming noproducing cost).

The price under |η| = 1 is the optimal price (if no producing cost)I Equivalent to the previous first-order condition.


Demand Elasticity

Now consider the production cost, where profit = revenue - cost.

Suppose the unit producing cost is C . Then, the optimal price isgiven by 4π = C · 4Q, or equivalently,

P · 4Q ·(

1− 1

|η|

)= C · 4Q.

Hence at the optimal price, we have

|η| =1

1− C/P> 1

Recall thatI When |η| > 1, we say that the demand curve is elastic.I When |η| < 1, we say that the demand curve is inelastic.

Theorem

A monopolist always operates on the elastic portion of the demand curve.


Demand Elasticity

When 4Q = 1, then

I 4π = P ·(

1− 1|η|

)is called the marginal revenue (MR);

I C is the marginal cost (MC). In general, MC may not be a constant.

Hence the optimal production quality equalizes MR and MC.


Section 5.2Theory: Price Discriminations


What is Price Discrimination?

Price discrimination (or price differentiation) is a pricing strategywhere products are transacted at different prices in different marketsor territories.

Examples of Price Discriminations:I Charge different prices to the same consumer, e.g., for different units of

products;I Charge uniform but different prices to different groups of consumers for

the same product.

Types of Price DiscriminationI First-degree price discriminationI Second-degree price discriminationI Third-degree price discrimination


An Illustrative Example

How would the monopolist increase his profit via price discrimination?

I MR: the marginal revenue curve;I MC: the marginal cost curve;I Demand: the downward-sloping demand curve;

Quantity

Price

0 Q∗ Q?

C0

P∗

MC

Demand

MR

π∗

π+

π?


Without Price Discrimination

Without price discrimination, the monopolist charges a singlemonopoly price to all consumers:

The optimal production quality (and demand) is Q∗, which equalizesMC and MR;

The optimal monopoly price is P∗, which is determined by the Q∗

and the demand curve.

The monopolist’s profit (=revenue - cost) is π∗, and the consumersurplus is π+.


With Price Discrimination

With price discrimination, the monopolist can charge different pricesto different consumers:

For example, the monopolist can charge each consumer the most thathe would be willing to pay for each product that he buys;

With the same demand Q∗, the monopolist’s profit is π∗ + π+, andthe consumer surplus is 0;

When the demand increases to Q?, the monopolist’s profit isπ∗ + π+ + π?, and the consumer surplus is 0;


First Degree Price Discrimination

With the first-degree price discrimination (or perfect pricediscrimination), the monopolist charges each consumer the most thathe would be willing to pay for each product that he buys.

The monopolist captures all the market surplus, and the consumergets zero surplus.

It requires that the monopolist knows exactly the maximum price thatevery consumer is willing to pay for each product, i.e., the fullknowledge about every consumer demand curve.


Illustration of First Degree Price Discrimination

The consumer is willing to pay a maximum price P1 for the firstproduct, P2 for the second product, and so on.

Under the first-degree price discrimination, the consumer is chargedby P1 for the first product, P2 for the second product, and so on.

The monopolist captures all the market surplus (shadow area).

Quantity

Price

0 1 2 3 4 5 6 7 8

P1P2P3P4P5

C

...

MC

Demand


Second Degree Price Discrimination

With the second-degree price discrimination (or declining blockpricing), the monopolist offers a bundle of prices to each consumer,with different prices for different blocks of units.

The second-degree price discrimination can be viewed as a limitedversion of the first-degree price discrimination (where a different priceis set for every different unit).

The second-degree price discrimination can be viewed as a generalizedversion of the monopoly pricing (as it degrades to the monopolypricing when the number of prices is one).


Illustration of Second Degree Price DiscriminationUnder this second-degree price discrimination, the monopolist offers abundle of prices {P1,P

∗,P2} with P1 > P∗ > P2.I P1 is the unit price for the first block (the first Q1 units) of products;I P∗ is the unit price for the second block (from Q1 to Q∗) of products;I P2 is the unit price for the third block (from Q∗ to Q2).I The monopolist’s profit is illustrated by the shadow area, and the

consumer surplus is δ1 + δ∗ + δ2.

Quantity

Price

0 Q1 Q∗ Q2

P2P∗P1

δ1

δ∗δ2

MC

Demand

MR


First vs. Second Degree Price DiscriminationUnder the second-degree price discrimination {P1,P

∗,P2}:I The monopolist’s profit is illustrated by the shadow area, and the

consumer surplus is δ1 + δ∗ + δ2.

Under the first-degree price discrimination:I The monopolist charges a different price D(Q) for each unit of product;I The monopolist captures all the market surplus (the shadow area +δ1 + δ∗ + δ2, and the consumer achieves zero surplus.

Quantity

Price

0 Q1 Q∗ Q2

P2P∗P1

δ1

δ∗δ2

MC

Demand

MR


Single Monopoly Pricing vs. Second Degree PriceDiscriminations

Under the second-degree price discrimination {P1,P∗,P2}:

I The monopolist’s profit is illustrated by the shadow area, and theconsumer surplus is δ1 + δ∗ + δ2.

Under the monopoly pricing without price discrimination:I The optimal monopoly price is P∗ and the demand is Q∗;I The monopolist’s profit is P∗ ·Q∗ −

∫ Q∗

0MC (Q)dQ, and the consumer

surplus is δ1 + δ∗ + (P1 − P∗) · Q1.

Quantity

Price

0 Q1 Q∗ Q2

P2P∗P1

δ1

δ∗δ2

MC

Demand

MR


Second Degree Price Discrimination

Comparison of different pricing strategies

I When there is a single price, the second-degree price discriminationdegrades to the monopoly pricing;

I When the price bundle curve approximates to the inverse demand curveP(Q), the second-degree price discrimination converges to thefirst-degree price discrimination.


Third Degree Price Discrimination

Limitation of First- and Second-Degree Price DiscriminationsI Needs the full or partial demand curve information of every individual

consumer, and benefits from this information by charging the consumerdifferent prices for different units of products.

How should the monopolist discriminates the prices, if he does notknow the detailed demand curve information of each individualconsumer, but knows from experience that different groups ofconsumers have different total demand curves?

→ Third-Degree Price Discrimination



With the third-degree price discrimination (or multi-market pricediscrimination), the monopolist specifies different prices for differentconsumer groups (with different total demand curves).

I Example: The Disney Park offers different ticket prices to three playergroups: children, adults, and elders.

Third-degree price discrimination usually occurs whenI the monopolist faces multiple identifiably groups of consumers with

different total demand curves;I the monopolist knows the total demand curve of every consumer group

(but not the individual demand curve of each consumer.

How to Identify Customers?


By Age


By Time

Kindle 2I 02/2009: $399I 07/2009: $299I 10/2009: $259I 06/2010: $189


Even More Dynamic


More Innovative Ways



Consider a simple scenario:I Two groups (markets) of consumers:I The total demand curve in each market i ∈ {1, 2} is Di (P);I The monopolist decides the price Pi for each market i .

Key problem: How should the monopolist set the prices {P1,P2} tomaximize his profit?

I Whether to charge the same price or different prices in differentmarkets (groups)?

I Which market should get the lower price if the monopolist chargesdifferent prices?

I What is the relationship between the prices of two markets?



The monopolist’s profit π(P1,P2) under prices {P1,P2} is

π(P1,P2) , P1 · Q1 + P2 · Q2 − C (Q1 + Q2)

The first-order condition:

∂π(P1,P2)

∂Pi= Qi + Pi ·

dQi

dPi− C ′(Q1 + Q2) · dQi

dPi= 0

I Qi , Di (Pi ) is the demand curve in market i ;I ηi ,

Pi

Qi

dQi

dPiis the price elasticity of demand in market i ;

I C ′(Q1 + Q2) is the marginal cost (MC) of the monopolist;



The optimality condition:

C ′(Q1 + Q2) = Pi + Qi ·dPi

dQi= Pi ·

(1− 1

|ηi |

)⇒ Under the optimal prices (P∗1 ,P

∗2 ), the marginal revenues (MR) in

all markets are identical, and are equal to the marginal cost (MC):

P∗1 ·(

1− 1

|η1|

)= P∗2 ·

(1− 1

|η2|

)I Pi ·

(1− 1

|ηi |)

is the marginal revenue (MR) of the monopolist in

market i ;



The optimal prices (P∗1 ,P∗2 ) satisfy

P∗1 ·(

1− 1

|η1|

)= P∗2 ·

(1− 1

|η2|

)I If |η1| 6= |η2|, then P∗1 6= P∗2 . That is, the monopolist will charge

different prices when two markets have different price elasticities.I If |η1| > |η2|, then P∗1 < P∗2 . That is, the market with the higher price

elasticity will get a lower optimal price.


Third Degree Price DiscriminationGraphic Interpretation of Optimal Prices (P∗1 ,P

∗2 )

I Di: the demand curve in market i ;I MRi: the marginal revenue curve in market i ;I MR (the blue curve): the overall marginal revenue curve (summing

MR1 and MR2 horizontally);I MC (the red curve): the marginal cost curve;

Quantity

Price

MC

MR

D1 D2MR1 MR2

Q1 + Q2Q1 Q2

C0

P∗1

P∗2



Graphic Interpretation of Optimal Prices (P∗1 ,P∗2 )

I Market 1: the demand is Q1, the marginal revenue equals C0;I Market 2: the demand is Q2, the marginal revenue equals C0;I Total market demand is Q1 + Q2, and the marginal cost is C0;I C0 is at the intersection of MC and MR curves.

Quantity

Price

MC

MR

D1 D2MR1 MR2

Q1 + Q2Q1 Q2

C0

P∗1

P∗2



Necessary conditions to make the third-degree price discriminationapplicable and profitable:

I Monopoly power: The firm must have the monopoly power to affectmarket price (there is no price discrimination in perfectly competitivemarkets).

I Market segmentation: The firm must be able to split the market intodifferent groups of consumers, and also be able to identify the type ofeach consumer.

I Elasticity of demand: The price elasticities of demand in differentmarkets are different.


Section 5.3: Cellular Network Pricing


Network Model

A cellular operator with B Hz of bandwidth

Sell bandwidth to multiple users


Two-Stage Decision Process

Stage I: (Operator pricing)

The operator decides price p and announces to users

Stage II: (Users’ demands)

Each user decides how much bandwidth b to request


User’s Spectrum Efficiency

h: a user’s (average) channel gain between him and the base station

P: a user’s transmission power density (per unit bandwidth)

θ: a user’s spectrum efficiency (data rate per unit bandwidth)

θ = log2(1 + SNR) = log2

(1 +

Ph

n0

)When allocated bandwidth b, the user achieves a data rate of θb


User’s Spectrum Efficiency

Different users have different spectrum efficienciesI Due to different values of P and hI Indoor users often have a smaller h than outdoor users

Normalize the range of θ to be [0,1]I Divided by the maximum value of θ among all users

0 1θ


User’s Utility and Payoff

A user’s utility when allocated bandwidth b

u(θ, b) = ln(1 + θb)

A user’s payoff under linear pricing p:

π(θ, b, p) = ln(1 + θb)− pb


User’s Demand in Stage II

Payoff maximization problem

maxb≥0

π(θ, b, p) = maxb≥0

(ln(1 + θb)− pb)

Concave maximization problem ⇒ user’s optimal demand

b∗(θ, p) =

{ 1p −

1θ , if p ≤ θ,

0, otherwise.

User’s maximum payoff

π(θ, b∗(θ, p), p) =

{ln(θp

)− 1 + p

θ , if p ≤ θ,

0, otherwise.


User Separation Based on Spectrum Efficiency

No service Cellular service

0 p 1θ


Users’ Total Demand

Price p ≤ maxθ∈[0,1] θ = 1I If p > 1, the total user demand will be 0

Total user demand

Q(p) =

∫ 1

p

(1

p− 1

θ

)dθ =

1

p− 1 + ln p

I Decreasing in p.


Operator’s Optimal Pricing

Operator’s revenue maximization problem

max0<p≤1

min (pB, pQ(p))

I pB is increasing in pI pQ(p) is decreasing in p:

dpQ(p)

dp= ln p < 0

I We can show that at the optimal price p∗, p∗B = p∗Q(p∗).

The optimal price p∗ is the unique solution of

B =1

p∗− 1 + ln p∗

I B → 0⇒ p → 1I B →∞⇒ p → 0


Section 5.4: Partial Price Differentiation


Network Model

One wireless service provider (SP)

A set of I groups of users, where each group i ∈ I hasI Ni homogenous usersI Same utility function ui (si ) = θi ln(1 + si )I Groups have decreasing preference coefficients: θ1 > θ2 > · · · > θI

The SP’s decision for each group iI Admit ni ≤ Ni usersI Charge a unit price pi (per unit of resource)I Subject to total resource limit:

∑i ni si ≤ S


Two-Stage Decision Process

Stage I: (Service provider’s pricing and admission control)

The SP decides price pi and ni for each group i

Stage II: (Users’ demands)

Each user in group i decides the demand si

Complete price differentiation: charge up to I different prices

Single pricing (no price differentiation): charge one price

Partial price differentiation: charge J prices with 1 ≤ J ≤ I


Complete Price Differentiation: Stage II

Each (admitted) group i user chooses si to maximize payoff

maximizesi≥0

(θi ln(1 + si )− pi si )

The unique optimal demand is

s∗i (pi ) = max

(θipi− 1, 0

)=

(θipi− 1

)+


Complete Price Differentiation: Stage I

SP performs admission control n and determines prices p:

maximizen,p≥0,s≥0

∑i∈I

nipi si

subject to si =

(θipi− 1

)+

, i ∈ I,

ni ∈ {0, . . . ,Ni} , i ∈ I,∑i∈I

ni si ≤ S .

I The Stage II’s user responses are incorporated

This problem is challenging to solve due to non-convex objectives,integer variables, and coupled constraint.


Complete Price Differentiation: Stage I

The admission control and pricing can be decoupled

At the unique optimal solution

I Do not reject any userI Charge prices such that users perform voluntary admission control:

there exists a group threshold K cp and λcp with

p∗i =

{ √θiλ∗, i ≤ K cp;

θi , i > K cp.

and

s∗i =

{ √θiλ∗ − 1, i ≤ K cp;

0, i > K cp.

I The choice of λ∗ satisfies

K cp∑i=1

ni

(√θiλ∗− 1

)= S


Complete Price Differentiation: Optimal Solution

706 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 22, NO. 3, JUNE 2014

where is the Lagrange multiplier corresponding to theresource constraint (9).

Meanwhile, we note the resource constraint (9) must holdwith equality since the objective is strictly increasing functionin . Thus, by plugging (11) into (9), we have

(12)

This weighted water-filling problem (where can be viewedas the water level) in general has no closed-form solution for. However, we can efficiently determine the optimal solutionby exploiting the special structure of our problem. Note

that since , then must satisfy the followingcondition:

(13)

for a group index threshold value satisfying

and (14)

In other words, only groups with index no larger than willbe allocated the positive resource. This property leads to thefollowing simple Algorithm 1 to compute and group indexthreshold : We start by assuming and compute .If (14) is not satisfied, we decrease by one and recomputeuntil (14) is satisfied.

Algorithm 1: Solving the Resource Allocation Problem

1: function

2: ,

3: while do

4: ,

5: end while6: ,7: return ,8: end function

Since , Algorithm 1 always con-verges and returns the unique values of and . The com-plexity is , i.e., linear in the number of user groups (not thenumber of users).With and , the solution of the resource allocation

problem can be written as

otherwise(15)

For the ease of discussions, we introduce a new notion ofthe effective market, which denotes all the groups allocatednonzero resource. For the resource allocation subproblem ,the threshold describes the size of the effective market.All groups with indices no larger than are effective groups,

Fig. 2. Six-group example for the effective market: The willingness to pay de-creases from group 1 to group 6. The effective market threshold can be obtainedby Algorithm 1 and is 4 in this example.

and users in these groups as effective users. An example of theeffective market is illustrated in Fig. 2.Now let us solve the admission control subproblem .

Denote the objective (10) as , by (15), then

.We first relax the integer do-

main constraint of as . Since (13), by taking thederivative of the objective function with respect to ,we have

(16)

Also from (13), we have , thus

, for , and ,for . This means that the objectiveis strictly increasing in for all , thus it is op-timal to admit all users in the effective market. The admissiondecisions for groups not in the effective market are irrelevantto the optimization since those users consume zero resource.Therefore, one of the optimal solutions of the subproblemis for all . Solving the and thesubproblems leads to the optimal solution of the problem.Theorem 1: There exists an optimal solution of the

problem that satisfies the following conditions.• All users are admitted: for all• There exist a value and a group index threshold

, such that only the top groups of users receive pos-itive resource allocations

otherwise

with the prices

otherwise.

The values of and can be computed as inAlgorithm 1 by setting , for all .

Theorem 1 provides the right economic intuition: The serviceprovider maximizes its revenue by charging a higher price tousers with a higher willingness to pay. It is easy to check that

for any . The low-willingness-to-pay users areexcluded from the markets.

C. Properties

Here, we summarize some interesting properties of the op-timal complete price differentiation scheme.

Effective market: includes groups receiving positive resources


Single Pricing (No Price Differentiation)

Problem formulation similar as the complete price differentiation case

Key difference: change the same price p to all groups

Similar optimal solution structure

Effective market is no larger than the one under complete pricedifferentiation

I Less users will be served


Effectiveness of Complete Price Differentiation

200 400 600 800 1000

0.05

0.10

0.15

G�Θ, N, S�

S

0

20

40

60

80

N1 N2 N3

N1 N2 N3

N1 N2 N3

Case 1

Case 2

Case 3

0

20

40

60

80

0

20

40

60

80

Relative Revenue Gain =RComplete Price Differentiation − RSingle Price

RSingle Price

Relative revenue gain of price differentiation reaches the maximum ifI The high willingness-to-pay users are minority, andI Total resource S is limited


Partial Price Differentiation

The most general case

SP can charge J prices to I groups, where J ≤ I

I Complete price differentiation: J = I

I Single pricing: J = 1

How to divide I groups into J clusters, and optimize the J prices?


Partial Price Differentiation

a={aji , j ∈ J , i ∈ I}: binary variables defining the partition

I aji = 1 ⇒ group i is in cluster j

Revenue optimization problem:

maximize{ni ,pi ,si ,pj ,aji }∀i,j

∑i∈I

nipi si

subject to si =

(θipi− 1

)+

, ∀ i ∈ I,

ni ∈ {0, . . . ,Ni}, ∀ i ∈ I,∑i∈I

ni si ≤ S ,

pi =∑j∈J

ajipj ,

∑j∈J

aji = 1, aji ∈ {0, 1}, ∀ i ∈ I.


Three-Level Decomposition

Level-1 (Cluster Partition): partition I groups into J clusters

Level-2 (Inter-Cluster Resource Allocation): allocate resources amongclusters (subject to the total resource constraint)

Level-3 (Intra-Cluster Pricing and Resource Allocation): optimizepricing and resource allocations within each cluster


Level 3: Pricing and Resource Allocation in SingleCluster

Given a fixed partition a and a cluster resource allocation s ∆= {s j}j∈J

Solve the pricing and resource allocation problems in cluster Cj :

Level-3: maximizeni ,si ,pj

∑i∈Cj

nipjsi

subject to si =

(θipj− 1

)+

, ∀ i ∈ Cj ,

ni ≤ Ni , ∀ i ∈ Cj ,∑i∈Cj

ni si ≤ s j .

Equivalent to a single pricing problem


Level 3: Effective Market in a Single Cluster

LI AND HUANG: PRICE DIFFERENTIATION FOR COMMUNICATION NETWORKS 709

TABLE INUMERICAL EXAMPLES FOR FEASIBLE SET SIZE OF THE PARTITION PROBLEM IN LEVEL 1

Fig. 5. Illustrative example of Level 3: The cluster contains fourgroups—groups 4, 5, 6 and 7—and the effective market contains groups4 and 5, thus .

The problem is a combinatorial optimization problemand is more difficult than the previous and problems.On the other hand, we notice that the problem formulationincludes the scheme and the scheme scenario

as special cases. The insights we obtained from solvingthese two special cases in Sections III and IV will be helpful tosolve the general problem.

A. Three-Level Decomposition

To solve the problem, we decompose and tackle it inthree levels. In the lowest Level 3, we determine the pricingand resource allocation for each cluster, given a fixed partitionand fixed resource allocation among clusters. In Level 2, wecompute the optimal resource allocation among clusters, givena fixed partition. In Level 1, we optimize the partition amonggroups.1) Level-3: Pricing and Resource Allocation in Each Cluster:

For a fixed partition and a cluster resource allocation, we focus the pricing and resource allocation problems

within each cluster ,

Level 3

subject to

The Level-3 subproblem coincides with the scheme dis-cussed in Section IV since all groups within the same clusterare charged with a single price .We can then directly apply theresults in Theorem 2 to solve the Level-3 problem. We denotethe effective market threshold2 for cluster as , which canbe computed in Algorithm 2. An illustrative example is shownin Fig. 5, where the cluster contains four groups (group 4, 5, 6,and 7), and the effective market contains groups 4 and 5, thus

2Note that we do not assume that the effective market threshold equals to thenumber of effective groups, e.g., there are two effective groups in Fig. 5, butthreshold . Later, we will prove that there is a unified threshold for the

problem. Then, by this result, the group index threshold actually coincideswith the number of effective groups.

. The service provider obtains the following maximumrevenue obtained from cluster :

(25)

2) Level-2: Resource Allocation Among Clusters: For afixed partition , we then consider the resource allocationamong clusters

Level 2

subject to

We will show in Section V-B that subproblems in Levels 2 and 3can be transformed into a complete price differentiation problemunder proper technique conditions. Let us denote the its optimalvalue as .3) Level-1: Cluster Partition: Finally, we solve the cluster

partition problem

Level 1

subject to

This partition problem is a combinatorial optimiza-tion problem. The size of its feasible set is

, Stirling number of the secondkind [27, Ch.13], where is the binomial coefficient.Some numerical examples are given in the third row in Table I.If the number of prices is given, the feasible set size isexponential in the total number of groups . For our problem,however, it is possible to reduce the size of the feasible setby exploiting the special problem structure. More specifically,the group indices in each cluster should be consecutive atthe optimum. This means that the size of the feasible set is

as shown in the last row in Table I, and thusis much smaller than .Next, we discuss how to solve the three level subproblems. A

route map for the whole solving process is given in Fig. 6.

B. Solving Levels 2 and 3

The optimal solution (25) of the Level-3 problem can beequivalently written as

(26)

where (27)


Level 2: Resource Allocation Among Clusters

For a fixed partition a

Consider the resource allocation among clusters:

Level-2: maximizes j≥0

∑j∈J

R j(s j , a)

subject to∑j∈J

s j ≤ S .

Solving Level 2 and Level 3 together is equivalent of solving acomplete price differentiation problem


Level-1: Cluster Partition

Level-1: maximizeaji∈{0,1},∀i ,j

Rpp(a)

subject to∑j∈J

aji = 1, i ∈ I.


How to Perform Cluster Partition in Level 1

Naive exhaustive search leads to formidable complexity for Level 1

Groups I = 10 I = 100 I = 1000

Clusters J = 2 J = 3 J = 2 J = 2

Combinations 511 9330 6.33825× 1029 5.35754× 10300

Do we need to check all partitions?


Property of An Optimal Partition

Will the following partition ever be optimal?

710 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 22, NO. 3, JUNE 2014

Fig. 6. Decomposition and simplification of the general problem: Thethree-level decomposition structure of the problem is shown in the left-handside. After simplifications in Sections V-B and V-C, the problem will be reducedto structure in right-hand side.

Fig. 7. Example of coupling thresholds.

The equality (a) in (26) means that each cluster can be equiv-alently treated as a group with homogeneous users with thesame willingness to pay . We name this equivalent group asa super-group (SG). We summarize the above result as the fol-lowing lemma.Lemma 1: For every cluster and total resource , ,

we can find an equivalent super-group that satisfies conditionsin (27) and achieves the same revenue under the scheme.Based on Lemma 1, the Level-2 and Level-3 subproblems

together can be viewed as the problem for super-groups.Since a cluster and its super-group form a one-to-one mapping,we will use the two words interchangeably in the sequel.However, simply combining Theorems 1 and 2 to solve the

Level-2 and Level-3 subproblems for a fixed partition can re-sult in a very high complexity. This is because the effective mar-kets within each super-group and between super-groups are cou-pled together. An illustrative example of this coupling effectivemarket is shown in Fig. 7, where is the threshold betweenclusters and has three possible positions (i.e., between groups 2and 3, between groups 5 and 6, or after group 6); and andare thresholds within cluster and , which have two or threepossible positions, respectively. Thus, there arepossible thresholds possibilities in total.The key idea to resolve this coupling issue is to show that

the situation in Fig. 7 cannot be an optimal solution of theproblem. The results in Sections III and IV show that there is aunified threshold at the optimum in both the and cases,e.g., Fig. 2. Next, we will show that a unified single thresholdalso exists in the case.Lemma 2: At any optimal solution of the scheme, the

group indices of the effective market are consecutive.The proof of Lemma 2 can be found in our online technical

report[25]. The intuition is that the resource should be always al-located to high-willingness-to-pay users at the optimum. Thus,it is not possible to have Fig. 7 at an optimal solution, where

high-willingness-to-pay users in group 2 are allocated zero re-source, while low-willingness-to-pay users in group 3 are allo-cated positive resources.Based on Lemma 2, we know that there is a unified effective

market threshold for the problem, denoted as . Since allgroups with indices larger than make zero contribution tothe revenue, we can ignore them and only consider the partitionproblem for the first groups. Given a partition that dividesthe groups into clusters (super-groups), we can applythe result in Section III to compute the optimal revenue inLevel-2 based on Theorem 1

(28)

C. Solving Level-1

1) With a Given Effective Market Threshold : Based onthe previous results, we first simplify the Level-1 subproblemand prove the theorem below.Theorem 3: For a given threshold , the optimal partition

of the Level-1 subproblem is the solution of the following opti-mization problem.

Level 1

subject to

(29)

where , is the value of averagewillingness to pay of the th group for the partition , and

Proof: The objective function and the first three con-straints in Level 1 are easy to understand: if the effectivemarket threshold is given, then the objective function ofthe Level-1 subproblem, maximizing in (28) over , is assimple as minimizing as the Level-1 problemsuggested; the first three constraints are given by the definitionof the partition.Constraint (29) is the threshold condition that supports (28),

which means that the least-willingness-to-pay users in the ef-fective market have a positive demand. It ensures that calcu-lating the revenue by (28) is valid. Remember that the solutionof the problem of Levels 2 and 3 is threshold-based, andLemma 2 indicates that (29) is sufficient for that all groups withwillingness larger than group can have positive demands.

No.

We prove that group indices in the effective market are consecutive.


Reduced Complexity of Cluster Partition in Level I

The search complexity reduces to polynomial in I .

Groups I = 10 I = 100 I = 1000

Clusters J = 2 J = 3 J = 2 J = 2

Combinations 511 9330 6.33825× 1029 5.35754× 10300

Reduced Combos 9 36 99 999


Relative Revenue Gain

10 20 30 40 50

0.02

0.04

0.06

0.08

0.10

0.12

0.14

100 200 300 400 500

0.05

0.10

0.15

0.20

GG

S

S

Complete price differentiation�Five Prices�Four Prices

Three Prices

Two Prices

A total of I = 5 groups

Plot the relative revenue gain of price differentiation vs. total resource

Maximum gains in the small plotI J = 3 is the sweet spot


Section 5.5: Chapter Summary


Key Concepts

TheoryI Monopoly pricing and the demand elasticityI First-degree price discriminationI Second-degree price discriminationI Third-degree price discrimination

ApplicationI Cellular Network PricingI Partial Price Discrimiation


References and Extended Reading

L. Duan, J. Huang, and B. Shou, “Economics of Femtocell ServiceProvisions,” IEEE Transactions on Mobile Computing, vol. 12, no. 11,pp. 2261 - 2273, November 2013

S. Li and J. Huang, “Price Differentiation for Communication Networks,”

IEEE Transactions on Networking, vol. 22, no. 2, pp. 703 - 716, June 2014




wireless network pricing chapter 5: monopoly and price...

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